AI 中文总结
本文研究Rⁿ开子集上的s-分数阶p-拉普拉斯型非局部非线性方程，明确了(s₁,p₁)正则性推出(s₂,p₂)正则性的条件，推导两种容量的比较估计，证明该框架下超调和函数的细连续性及极集与零容量集的一致性。
AI 中文摘要
本文研究欧氏空间Rⁿ开子集上的s-分数阶p-拉普拉斯型非局部非线性方程，探讨解的边界正则性如何依赖参数s与p，包括局部情形s=1。具体而言，本文明确了(s₁,p₁)对应的正则性何时能推出(s₂,p₂)对应的正则性，证明依赖于用 condenser 容量与 Sobolev 容量表述的 Wiener 准则的等价性；为建立该等价性，本文推导了两种容量间的精确比较估计。Wiener积分定义了细薄性与细拓扑，本文证明与非局部非线性算子关联的每个超调和函数都是细连续的，还证明该分数阶框架下的极集与零容量集一致。
英文摘要
In this paper we study nonlocal nonlinear equations of $s$-fractional $p$-Laplacian type in open subsets of $\mathbf{R}^n$. We investigate how the boundary regularity of solutions depends on the parameters $s$ and $p$, including the local case $s=1$. Specifically, we show exactly when regularity for $(s_1, p_1)$ implies regularity for $(s_2, p_2)$. The proof relies on the equivalence between Wiener criteria formulated with condenser and Sobolev capacities. To establish this equivalence, we derive precise comparison estimates between the two capacities. The Wiener integral defines thinness and the fine topology. We show that every superharmonic function associated with a nonlocal nonlinear operator is finely continuous. Moreover, we prove that polar sets in this fractional setting coincide with sets of zero capacity.
